So I am given a 2-dimensional harmonic oscillator with $H=H_1+H_2$ where
$$H_i=\frac{p_i^2}{2m}+\frac{1}{2}m\omega^2x_i^2$$
Additionally,
$$L=x_1p_2-x_2p_1$$
If we define
$$A=\frac{1}{2\omega}[H_1-H_2]$$
$$B=\frac{1}{2}L$$
$$C=\frac{-i}{\hbar}[A,B]$$
Where [A,B] is the commuatator of A with B. We are asked for the explicit form of C, but isnt it just
$$[H_1-H_2,L] = [H_1,L]-[H_2,L]=0$$
Due to the isotropy of space. It just does not make sense that C would be 0, because then the three would not be closed under commutation (which I am supposed to show).